3.23.77 \(\int \frac {1}{\sqrt [3]{-x+x^3} (1+x^6)} \, dx\) [2277]

Optimal. Leaf size=173 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-x+x^3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-x+x^3}\right )}{6 \sqrt [3]{2}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-x+x^3}+\sqrt [3]{2} \left (-x+x^3\right )^{2/3}\right )}{12 \sqrt [3]{2}}-\frac {1}{6} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {-\log (x)+\log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ] \]

[Out]

Unintegrable

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Rubi [C] Result contains complex when optimal does not.
time = 0.76, antiderivative size = 1276, normalized size of antiderivative = 7.38, number of steps used = 25, number of rules used = 16, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.842, Rules used = {2081, 6847, 2099, 2174, 2183, 384, 502, 206, 31, 648, 631, 210, 642, 455, 58, 6860} \begin {gather*} \frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \text {ArcTan}\left (\frac {1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{x^2-1}}}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3-x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{2} x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{3 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{x^2-1} \text {ArcTan}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \text {ArcTan}\left (\frac {1-2^{2/3} \sqrt [3]{x^2-1}}{\sqrt {3}}\right )}{6 \sqrt [3]{2} \sqrt {3} \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (-\left (\left (1-x^{2/3}\right ) \left (x^{2/3}+1\right )^2\right )\right )}{36 \sqrt [3]{2} \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (-2 x^2-i \sqrt {3}+1\right )}{12 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (-2 x^2+i \sqrt {3}+1\right )}{12 \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (x^2+1\right )}{36 \sqrt [3]{2} \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{x^2-1}}\right )}{18 \sqrt [3]{2} \sqrt [3]{x^3-x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (\frac {2^{2/3} \left (1-x^{2/3}\right )^2}{\left (x^2-1\right )^{2/3}}+\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{x^2-1}}+1\right )}{36 \sqrt [3]{2} \sqrt [3]{x^3-x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (\sqrt [3]{2} x^{2/3}-\sqrt [3]{x^2-1}\right )}{6 \sqrt [3]{2} \sqrt [3]{x^3-x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x^{2/3}-\sqrt [3]{x^2-1}\right )}{4 \sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x^3-x}}+\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (\sqrt [3]{x^2-1}+\sqrt [3]{2}\right )}{12 \sqrt [3]{2} \sqrt [3]{x^3-x}}-\frac {\sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (-x^{2/3}+2^{2/3} \sqrt [3]{x^2-1}+1\right )}{12 \sqrt [3]{2} \sqrt [3]{x^3-x}}-\frac {\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x} \sqrt [3]{x^2-1} \log \left (x^{2/3}-\sqrt [3]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [3]{x^2-1}\right )}{4 \sqrt [3]{x^3-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((-x + x^3)^(1/3)*(1 + x^6)),x]

[Out]

(x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 - (2^(1/3)*(1 - x^(2/3)))/(-1 + x^2)^(1/3))/Sqrt[3]])/(6*2^(1/3)*Sqrt[3]*(
-x + x^3)^(1/3)) - (x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 + (2*2^(1/3)*(1 - x^(2/3)))/(-1 + x^2)^(1/3))/Sqrt[3]])
/(6*2^(1/3)*Sqrt[3]*(-x + x^3)^(1/3)) + (x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 + (2*2^(1/3)*x^(2/3))/(-1 + x^2)^(
1/3))/Sqrt[3]])/(3*2^(1/3)*Sqrt[3]*(-x + x^3)^(1/3)) + ((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*x^(1/3)*(-1 + x
^2)^(1/3)*ArcTan[(1 + (2*x^(2/3))/((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*(-1 + x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt
[3]*(-x + x^3)^(1/3)) + (x^(1/3)*(-1 + x^2)^(1/3)*ArcTan[(1 + (2*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*x^(2/3
))/(-1 + x^2)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*(-x + x^3)^(1/3)) + (x^(1/3)*
(-1 + x^2)^(1/3)*ArcTan[(1 - 2^(2/3)*(-1 + x^2)^(1/3))/Sqrt[3]])/(6*2^(1/3)*Sqrt[3]*(-x + x^3)^(1/3)) + (x^(1/
3)*(-1 + x^2)^(1/3)*Log[-((1 - x^(2/3))*(1 + x^(2/3))^2)])/(36*2^(1/3)*(-x + x^3)^(1/3)) + (x^(1/3)*(-1 + x^2)
^(1/3)*Log[1 - I*Sqrt[3] - 2*x^2])/(12*(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*(-x + x^3)^(1/3)) + ((-((I - Sqr
t[3])/(I + Sqrt[3])))^(1/3)*x^(1/3)*(-1 + x^2)^(1/3)*Log[1 + I*Sqrt[3] - 2*x^2])/(12*(-x + x^3)^(1/3)) + (x^(1
/3)*(-1 + x^2)^(1/3)*Log[1 + x^2])/(36*2^(1/3)*(-x + x^3)^(1/3)) + (x^(1/3)*(-1 + x^2)^(1/3)*Log[1 - (2^(1/3)*
(1 - x^(2/3)))/(-1 + x^2)^(1/3)])/(18*2^(1/3)*(-x + x^3)^(1/3)) - (x^(1/3)*(-1 + x^2)^(1/3)*Log[1 + (2^(2/3)*(
1 - x^(2/3))^2)/(-1 + x^2)^(2/3) + (2^(1/3)*(1 - x^(2/3)))/(-1 + x^2)^(1/3)])/(36*2^(1/3)*(-x + x^3)^(1/3)) -
(x^(1/3)*(-1 + x^2)^(1/3)*Log[2^(1/3)*x^(2/3) - (-1 + x^2)^(1/3)])/(6*2^(1/3)*(-x + x^3)^(1/3)) - (x^(1/3)*(-1
+ x^2)^(1/3)*Log[(-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*x^(2/3) - (-1 + x^2)^(1/3)])/(4*(-((I - Sqrt[3])/(I +
Sqrt[3])))^(1/3)*(-x + x^3)^(1/3)) + (x^(1/3)*(-1 + x^2)^(1/3)*Log[2^(1/3) + (-1 + x^2)^(1/3)])/(12*2^(1/3)*(
-x + x^3)^(1/3)) - (x^(1/3)*(-1 + x^2)^(1/3)*Log[1 - x^(2/3) + 2^(2/3)*(-1 + x^2)^(1/3)])/(12*2^(1/3)*(-x + x^
3)^(1/3)) - ((-((I - Sqrt[3])/(I + Sqrt[3])))^(1/3)*x^(1/3)*(-1 + x^2)^(1/3)*Log[x^(2/3) - (-((I - Sqrt[3])/(I
+ Sqrt[3])))^(1/3)*(-1 + x^2)^(1/3)])/(4*(-x + x^3)^(1/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-(b*c - a*d)/b, 3]}, Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && NegQ
[(b*c - a*d)/b]

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 502

Int[(x_)/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[b/a, 3]}, Dist[-q^2/(3
*d), Int[1/((1 - q*x)*(a + b*x^3)^(1/3)), x], x] + Dist[q/d, Subst[Int[1/(1 + 2*a*x^3), x], x, (1 + q*x)/(a +
b*x^3)^(1/3)], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + a*d, 0]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 2099

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rule 2174

Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^3)^(1/3)), x_Symbol] :> Simp[Sqrt[3]*(ArcTan[(1 - 2^(1/3)*Rt[b,
3]*((c - d*x)/(d*(a + b*x^3)^(1/3))))/Sqrt[3]]/(2^(4/3)*Rt[b, 3]*c)), x] + (Simp[Log[(c + d*x)^2*(c - d*x)]/(2
^(7/3)*Rt[b, 3]*c), x] - Simp[(3*Log[Rt[b, 3]*(c - d*x) + 2^(2/3)*d*(a + b*x^3)^(1/3)])/(2^(7/3)*Rt[b, 3]*c),
x]) /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^3 + a*d^3, 0]

Rule 2183

Int[(Px_.)*((c_) + (d_.)*(x_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^3)^(p_.), x_Symbol] :> Dist[1/c^q, Int[E
xpandIntegrand[(c^3 - d^3*x^3)^q*(a + b*x^3)^p, Px/(c - d*x)^q, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] &&
PolyQ[Px, x] && EqQ[d^2 - c*e, 0] && ILtQ[q, 0] && RationalQ[p] && EqQ[Denominator[p], 3]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [3]{-x+x^3} \left (1+x^6\right )} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-1+x^2} \left (1+x^6\right )} \, dx}{\sqrt [3]{-x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3} \left (1+x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{9 (1+x) \sqrt [3]{-1+x^3}}+\frac {2-x}{9 \left (1-x+x^2\right ) \sqrt [3]{-1+x^3}}+\frac {2-x^3}{3 \sqrt [3]{-1+x^3} \left (1-x^3+x^6\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{6 \sqrt [3]{-x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {2-x}{\left (1-x+x^2\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{6 \sqrt [3]{-x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {2-x^3}{\sqrt [3]{-1+x^3} \left (1-x^3+x^6\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (\left (1-x^{2/3}\right ) \left (1+x^{2/3}\right )^2\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {-1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^3}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^3}}\right ) \, dx,x,x^{2/3}\right )}{6 \sqrt [3]{-x+x^3}}+\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \left (\frac {-1-i \sqrt {3}}{\sqrt [3]{-1+x^3} \left (-1-i \sqrt {3}+2 x^3\right )}+\frac {-1+i \sqrt {3}}{\sqrt [3]{-1+x^3} \left (-1+i \sqrt {3}+2 x^3\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (\left (1-x^{2/3}\right ) \left (1+x^{2/3}\right )^2\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{6 \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3} \left (-1-i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{-1+x^3}} \, dx,x,x^{2/3}\right )}{6 \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3} \left (-1+i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1-i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1+i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (i-\sqrt {3}-2 i x^{2/3}\right )^2 \left (i-\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (i+\sqrt {3}-2 i x^{2/3}\right )^2 \left (i+\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (\left (1-x^{2/3}\right ) \left (1+x^{2/3}\right )^2\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1+i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{-1-i \sqrt {3}-\left (1-i \sqrt {3}\right ) x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{-1+i \sqrt {3}-\left (1+i \sqrt {3}\right ) x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{2 \sqrt [3]{-x+x^3}}\\ &=\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1-i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1+i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (i-\sqrt {3}-2 i x^{2/3}\right )^2 \left (i-\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (i+\sqrt {3}-2 i x^{2/3}\right )^2 \left (i+\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (\left (1-x^{2/3}\right ) \left (1+x^{2/3}\right )^2\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1+i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{-\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{1+i \sqrt {3}} x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {-2 \sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{1+i \sqrt {3}} x}{\left (1-i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1+i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{-\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{1-i \sqrt {3}} x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {-2 \sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{1-i \sqrt {3}} x}{\left (1+i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1-i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{-x+x^3}}\\ &=\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1-i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1+i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (i-\sqrt {3}-2 i x^{2/3}\right )^2 \left (i-\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (i+\sqrt {3}-2 i x^{2/3}\right )^2 \left (i+\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (\left (1-x^{2/3}\right ) \left (1+x^{2/3}\right )^2\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\sqrt [3]{1+i \sqrt {3}}+\frac {\sqrt [3]{1-i \sqrt {3}} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\sqrt [3]{1-i \sqrt {3}}+\frac {\sqrt [3]{1+i \sqrt {3}} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1+i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1+i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}}+2 \left (1-i \sqrt {3}\right )^{2/3} x}{\left (1+i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1-i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{12 \sqrt [3]{1-i \sqrt {3}} \left (1+i \sqrt {3}\right )^{2/3} \sqrt [3]{-x+x^3}}-\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1-i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{4 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}+\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}}+2 \left (1+i \sqrt {3}\right )^{2/3} x}{\left (1-i \sqrt {3}\right )^{2/3}-\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+i \sqrt {3}} x+\left (1+i \sqrt {3}\right )^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{12 \left (1-i \sqrt {3}\right )^{2/3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}\\ &=\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1-i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1+i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (i-\sqrt {3}-2 i x^{2/3}\right )^2 \left (i-\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (i+\sqrt {3}-2 i x^{2/3}\right )^2 \left (i+\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (\left (1-x^{2/3}\right ) \left (1+x^{2/3}\right )^2\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\frac {\left (1-i \sqrt {3}\right )^{2/3} x^{4/3}}{\left (-1+x^2\right )^{2/3}}-\frac {2^{2/3} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{12 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\frac {\left (1+i \sqrt {3}\right )^{2/3} x^{4/3}}{\left (-1+x^2\right )^{2/3}}-\frac {2^{2/3} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{12 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\sqrt [3]{1+i \sqrt {3}}+\frac {\sqrt [3]{1-i \sqrt {3}} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\sqrt [3]{1-i \sqrt {3}}+\frac {\sqrt [3]{1+i \sqrt {3}} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1+i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{1+i \sqrt {3}} x^{2/3}}{\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{-1+x^2}}\right )}{2\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{-x+x^3}}-\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{1-i \sqrt {3}} x^{2/3}}{\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-1+x^2}}\right )}{2\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}\\ &=\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} \left (1-x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1-i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {2+\frac {\sqrt [3]{2} \left (1+i \sqrt {3}+2 x^{2/3}\right )}{\sqrt [3]{-1+x^2}}}{2 \sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3} \sqrt [3]{-x+x^3}}+\frac {\left (3 i-\sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {2 x^{2/3}}{\sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{6\ 2^{2/3} \sqrt [3]{1-i \sqrt {3}} \sqrt [3]{-x+x^3}}-\frac {\left (3 i+\sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^2} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{6\ 2^{2/3} \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (i-\sqrt {3}-2 i x^{2/3}\right )^2 \left (i-\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (i+\sqrt {3}-2 i x^{2/3}\right )^2 \left (i+\sqrt {3}+2 i x^{2/3}\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (-\left (\left (1-x^{2/3}\right ) \left (1+x^{2/3}\right )^2\right )\right )}{24 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (1+i \sqrt {3}\right )^{2/3}+\frac {\left (1-i \sqrt {3}\right )^{2/3} x^{4/3}}{\left (-1+x^2\right )^{2/3}}-\frac {2^{2/3} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{12 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\left (1-i \sqrt {3}\right )^{2/3}+\frac {\left (1+i \sqrt {3}\right )^{2/3} x^{4/3}}{\left (-1+x^2\right )^{2/3}}-\frac {2^{2/3} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{12 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\sqrt [3]{1+i \sqrt {3}}+\frac {\sqrt [3]{1-i \sqrt {3}} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \sqrt [3]{\frac {i+\sqrt {3}}{i-\sqrt {3}}} \sqrt [3]{-x+x^3}}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (\sqrt [3]{1-i \sqrt {3}}+\frac {\sqrt [3]{1+i \sqrt {3}} x^{2/3}}{\sqrt [3]{-1+x^2}}\right )}{6 \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1+i \sqrt {3}+2 x^{2/3}-2\ 2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \log \left (1-x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )}{8 \sqrt [3]{2} \sqrt [3]{-x+x^3}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 200, normalized size = 1.16 \begin {gather*} \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2} \left (2^{2/3} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}}\right )-2 \log \left (-2 x^{2/3}+2^{2/3} \sqrt [3]{-1+x^2}\right )+\log \left (2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{-1+x^2}+\sqrt [3]{2} \left (-1+x^2\right )^{2/3}\right )\right )-4 \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x^2}-x^{2/3} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{24 \sqrt [3]{x \left (-1+x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((-x + x^3)^(1/3)*(1 + x^6)),x]

[Out]

(x^(1/3)*(-1 + x^2)^(1/3)*(2^(2/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2^(2/3)*(-1 + x^2)^(1/3))] -
2*Log[-2*x^(2/3) + 2^(2/3)*(-1 + x^2)^(1/3)] + Log[2*x^(4/3) + 2^(2/3)*x^(2/3)*(-1 + x^2)^(1/3) + 2^(1/3)*(-1
+ x^2)^(2/3)]) - 4*RootSum[1 - #1^3 + #1^6 & , (-2*Log[x^(1/3)] + Log[(-1 + x^2)^(1/3) - x^(2/3)*#1])/#1 & ])
)/(24*(x*(-1 + x^2))^(1/3))

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x^{3}-x \right )^{\frac {1}{3}} \left (x^{6}+1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3-x)^(1/3)/(x^6+1),x)

[Out]

int(1/(x^3-x)^(1/3)/(x^6+1),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^3-x)^(1/3)/(x^6+1),x, algorithm="maxima")

[Out]

integrate(1/((x^6 + 1)*(x^3 - x)^(1/3)), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^3-x)^(1/3)/(x^6+1),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**3-x)**(1/3)/(x**6+1),x)

[Out]

Integral(1/((x*(x - 1)*(x + 1))**(1/3)*(x**2 + 1)*(x**4 - x**2 + 1)), x)

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Giac [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.45, size = 967, normalized size = 5.59 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^3-x)^(1/3)/(x^6+1),x, algorithm="giac")

[Out]

-1/12*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-1/x^2 + 1)^(1/3))) + 1/6*(sqrt(3)*cos(4/9*pi)^
5 - 10*sqrt(3)*cos(4/9*pi)^3*sin(4/9*pi)^2 + 5*sqrt(3)*cos(4/9*pi)*sin(4/9*pi)^4 - 5*cos(4/9*pi)^4*sin(4/9*pi)
+ 10*cos(4/9*pi)^2*sin(4/9*pi)^3 - sin(4/9*pi)^5 + sqrt(3)*cos(4/9*pi)^2 - sqrt(3)*sin(4/9*pi)^2 - 2*cos(4/9*
pi)*sin(4/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos(4/9*pi) + 2*(-1/x^2 + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(4
/9*pi))) + 1/6*(sqrt(3)*cos(2/9*pi)^5 - 10*sqrt(3)*cos(2/9*pi)^3*sin(2/9*pi)^2 + 5*sqrt(3)*cos(2/9*pi)*sin(2/9
*pi)^4 - 5*cos(2/9*pi)^4*sin(2/9*pi) + 10*cos(2/9*pi)^2*sin(2/9*pi)^3 - sin(2/9*pi)^5 + sqrt(3)*cos(2/9*pi)^2
- sqrt(3)*sin(2/9*pi)^2 - 2*cos(2/9*pi)*sin(2/9*pi))*arctan(1/2*((-I*sqrt(3) - 1)*cos(2/9*pi) + 2*(-1/x^2 + 1)
^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(2/9*pi))) - 1/6*(sqrt(3)*cos(1/9*pi)^5 - 10*sqrt(3)*cos(1/9*pi)^3*sin(1/9*p
i)^2 + 5*sqrt(3)*cos(1/9*pi)*sin(1/9*pi)^4 + 5*cos(1/9*pi)^4*sin(1/9*pi) - 10*cos(1/9*pi)^2*sin(1/9*pi)^3 + si
n(1/9*pi)^5 - sqrt(3)*cos(1/9*pi)^2 + sqrt(3)*sin(1/9*pi)^2 - 2*cos(1/9*pi)*sin(1/9*pi))*arctan(-1/2*((-I*sqrt
(3) - 1)*cos(1/9*pi) - 2*(-1/x^2 + 1)^(1/3))/((1/2*I*sqrt(3) + 1/2)*sin(1/9*pi))) + 1/12*(5*sqrt(3)*cos(4/9*pi
)^4*sin(4/9*pi) - 10*sqrt(3)*cos(4/9*pi)^2*sin(4/9*pi)^3 + sqrt(3)*sin(4/9*pi)^5 + cos(4/9*pi)^5 - 10*cos(4/9*
pi)^3*sin(4/9*pi)^2 + 5*cos(4/9*pi)*sin(4/9*pi)^4 + 2*sqrt(3)*cos(4/9*pi)*sin(4/9*pi) + cos(4/9*pi)^2 - sin(4/
9*pi)^2)*log((-I*sqrt(3)*cos(4/9*pi) - cos(4/9*pi))*(-1/x^2 + 1)^(1/3) + (-1/x^2 + 1)^(2/3) + 1) + 1/12*(5*sqr
t(3)*cos(2/9*pi)^4*sin(2/9*pi) - 10*sqrt(3)*cos(2/9*pi)^2*sin(2/9*pi)^3 + sqrt(3)*sin(2/9*pi)^5 + cos(2/9*pi)^
5 - 10*cos(2/9*pi)^3*sin(2/9*pi)^2 + 5*cos(2/9*pi)*sin(2/9*pi)^4 + 2*sqrt(3)*cos(2/9*pi)*sin(2/9*pi) + cos(2/9
*pi)^2 - sin(2/9*pi)^2)*log((-I*sqrt(3)*cos(2/9*pi) - cos(2/9*pi))*(-1/x^2 + 1)^(1/3) + (-1/x^2 + 1)^(2/3) + 1
) + 1/12*(5*sqrt(3)*cos(1/9*pi)^4*sin(1/9*pi) - 10*sqrt(3)*cos(1/9*pi)^2*sin(1/9*pi)^3 + sqrt(3)*sin(1/9*pi)^5
- cos(1/9*pi)^5 + 10*cos(1/9*pi)^3*sin(1/9*pi)^2 - 5*cos(1/9*pi)*sin(1/9*pi)^4 - 2*sqrt(3)*cos(1/9*pi)*sin(1/
9*pi) + cos(1/9*pi)^2 - sin(1/9*pi)^2)*log((I*sqrt(3)*cos(1/9*pi) + cos(1/9*pi))*(-1/x^2 + 1)^(1/3) + (-1/x^2
+ 1)^(2/3) + 1) + 1/24*2^(2/3)*log(2^(2/3) + 2^(1/3)*(-1/x^2 + 1)^(1/3) + (-1/x^2 + 1)^(2/3)) - 1/12*2^(2/3)*l
og(abs(-2^(1/3) + (-1/x^2 + 1)^(1/3)))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (x^3-x\right )}^{1/3}\,\left (x^6+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x^3 - x)^(1/3)*(x^6 + 1)),x)

[Out]

int(1/((x^3 - x)^(1/3)*(x^6 + 1)), x)

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